The first law of thermodynamics simply states that energy is conserved. But it is useful to look at those two non-state variables work and heat. Both are changes in energy of a system, so we can write the first law as $$\Delta U=Q+W$$ where $U$ is the internal energy of the system, $Q$ is the energy added to the system by heating, and $W$ is the work done by the system (or the energy removed from the system by working). It is more convenient mathematically, however, to have a framework for talking about infinitesimal changes in the total energy… $$dU=đQ+đW$$ If I stretch a rubber band and snap it shut repeatedly, what will happen to its temperature? If students look puzzled, suggest they assume that as long as you do this reasonably quickly, the air doesn't get heated up much (nor the fingers).[SWBQ]

SWBQ: If you stretch and then let a rubber band snap, what will happen to the internal energy and the temperature of the rubber band?

This is an interesting example, as rubber bands may change temperature in either direction as it is being stretched, dependent on the type of rubber band used. However, because the process returns to its original state, you can use the First Law to show that the temperature must increase regardless.

In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

Money is also a nice quantity because it is conserved—just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.\footnote{Yes, this is ridiculous. It would be slightly less ridiculous if we were talking about nations and commodities, but also far less humorous.}

Thus your savings $S$ can be considered to be a function of your bagels $B$ and coffee $C$. In this problem we will also discuss the prices $P_B$ and $P_C$, which you may not assume are independent of $B$ and $C$. It may help to imagine that you have

The prices of bagels and coffee $P_B$ and $P_C$ have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of $P_C$ and $P_B$?

Write down the total differential of your savings, in terms of $B$, $C$, $P_B$ and $P_C$.

Use the equality of mixed partial derivatives (Clairut's theorem) to find a relationship between $P_B$, $P_C$, $B$ and $C$. Write this relationship mathematically, and also describe in words what it means.

Solve for the total differential of your net worth. Once again use Clairut's theorem considering second derivatives of $W$ to find a different partial derivative relationship between $P_B$, $P_C$, $B$ and $C$.

(ExtensiveIntensiveChecking) What goes here?

For each of the following equations, check whether it could possibly make sense. You will need to check both dimensions and whether the quantities involved are intensive or extensive. For each equation, explain your reasoning.

You may assume that quantities with subscripts such as $V_0$ have the same dimensions and intensiveness/extensiveness as they would have without the subscripts.

\[p = \frac{N^2k_BT}{V}\]

\[p = \frac{Nk_BT}{V}\]

\[U = \frac32 k_BT\]

\[U = - Nk_BT \ln\frac{V}{V_0}\]

\[S = - k_B \ln\frac{V}{V_0}\]

\[S = - k_B \ln\frac{V}{N}\]

(ExtensiveInternalEnergy) What goes here?

Consider a system which has an internal energy defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?

For each of the following equations, check whether it could possibly make sense. You will need to check both dimensions and whether the quantities involved are intensive or extensive. For each equation, explain your reasoning.

You may assume that quantities with subscripts such as $V_0$ have the same dimensions and intensiveness/extensiveness as they would have without the subscripts.